Question
Solve the following quadratic equation using square roots: −9+(x−3)^2/−4 = −4 .(1 point)
Responses
{28,−22}
left brace 28 comma negative 22 right brace
{2,−8}
left brace 2 comma negative 8 right brace
{8,−2}
left brace 8 comma negative 2 right brace
{16,−10}
Responses
{28,−22}
left brace 28 comma negative 22 right brace
{2,−8}
left brace 2 comma negative 8 right brace
{8,−2}
left brace 8 comma negative 2 right brace
{16,−10}
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \(-9 + \frac{(x - 3)^2}{-4} = -4\), we can follow these steps:
1. **Isolate the quadratic term**:
First, let's start by adding 9 to both sides of the equation:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
Simplifying the right side:
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by -4**:
To eliminate the denominator, we'll multiply both sides by -4:
\[
(x - 3)^2 = -20
\]
3. **Solve for the square root**:
Since we have a squared term equal to a negative number, this indicates that there is no real solution. However, we can express the potential solutions using imaginary numbers:
\[
(x - 3)^2 = 20 \implies x - 3 = \pm \sqrt{20}
\]
Since \(\sqrt{20} = 2\sqrt{5}\), we get:
\[
x - 3 = \pm 2\sqrt{5}
\]
Thus, we have:
\[
x = 3 + 2\sqrt{5} \quad \text{or} \quad x = 3 - 2\sqrt{5}
\]
4. **Approximate values** (if needed):
If you want a numerical approximation of these solutions:
- \(3 + 2\sqrt{5} \approx 3 + 4.472 = 7.472\)
- \(3 - 2\sqrt{5} \approx 3 - 4.472 = -1.472\)
However, since \((x - 3)^2 = -20\) gives rise to complex solutions, the correct solutions involve imaginary numbers. Therefore, given the options provided (which seem to pertain to real solutions), none of them are correct as we find no real solutions for this equation.
If you are looking for the final values of solutions as \(3 + 2\sqrt{5}\) and \(3 - 2\sqrt{5}\), we will not be able to match them to any of the provided responses \(\{28,−22\}, \{2,−8\}, \{8,−2\}, \{16,−10\}\) since they are not correct as per the problem's real solutions. Thus, this needs to be acknowledged.
If you require help with further specific solutions in real numbers or complex numbers or a different equation, feel free to ask!
1. **Isolate the quadratic term**:
First, let's start by adding 9 to both sides of the equation:
\[
\frac{(x - 3)^2}{-4} = -4 + 9
\]
Simplifying the right side:
\[
\frac{(x - 3)^2}{-4} = 5
\]
2. **Multiply both sides by -4**:
To eliminate the denominator, we'll multiply both sides by -4:
\[
(x - 3)^2 = -20
\]
3. **Solve for the square root**:
Since we have a squared term equal to a negative number, this indicates that there is no real solution. However, we can express the potential solutions using imaginary numbers:
\[
(x - 3)^2 = 20 \implies x - 3 = \pm \sqrt{20}
\]
Since \(\sqrt{20} = 2\sqrt{5}\), we get:
\[
x - 3 = \pm 2\sqrt{5}
\]
Thus, we have:
\[
x = 3 + 2\sqrt{5} \quad \text{or} \quad x = 3 - 2\sqrt{5}
\]
4. **Approximate values** (if needed):
If you want a numerical approximation of these solutions:
- \(3 + 2\sqrt{5} \approx 3 + 4.472 = 7.472\)
- \(3 - 2\sqrt{5} \approx 3 - 4.472 = -1.472\)
However, since \((x - 3)^2 = -20\) gives rise to complex solutions, the correct solutions involve imaginary numbers. Therefore, given the options provided (which seem to pertain to real solutions), none of them are correct as we find no real solutions for this equation.
If you are looking for the final values of solutions as \(3 + 2\sqrt{5}\) and \(3 - 2\sqrt{5}\), we will not be able to match them to any of the provided responses \(\{28,−22\}, \{2,−8\}, \{8,−2\}, \{16,−10\}\) since they are not correct as per the problem's real solutions. Thus, this needs to be acknowledged.
If you require help with further specific solutions in real numbers or complex numbers or a different equation, feel free to ask!